2017/09/27 by Martin Bachratý, Bachratý, Martin, Jana Šiagiová +3
Computer Science · Engineering · Mathematics · #05C12 #05E18 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems #math.CO #msc:05C12 #msc:05E18
paper · pdf · doi:10.48550/arxiv.1709.09760
12 pages
arxiv created 2017/09/27 · openalex publication_date 2017/09/27 · arxiv updated 2017/09/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The largest order n(d,k) of a graph of maximum degree d and diameter k cannot exceed the Moore bound, which has the form M(d,k)=dk - O(dk-1) for d→∞ and any fixed k. Known results in finite geometries on generalised (k+1)-gons imply, for k=2,3,5, the existence of an infinite sequence of values of d such that n(d,k)=dk - o(dk). This shows that for k=2,3,5 the Moore bound can be asymptotically approached in the sense that n(d,k)/M(d,k)→ 1 as d→∞; moreover, no such result is known for any other value of k≥ 2. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result. The second and the third author (2012) proved by a direct construction that the Moore bound for diameter k=2 can be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construction can be derived from generalised triangles with polarity. By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, we prove that for an infinite set of values of d there exist Cayley graphs of degree d, diameter 3, and order d3-O(d2.5). The Moore bound for diameter 3 can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter 5 from generalised hexagons with polarity.